Properties of Triangles Unit 6

5.5 Inequalities in One Triangle Lesson Goals State Standards for Geometry Use triangle measurements to decide which side is longest or which angle is largest. Apply the Triangle Inequality Theorem to determine if 3 lengths can form a triangle, and find possible lengths of a 3rd side given two. Know and use the Triangle Inequality Theorem. Prove relationships between angles in a polygon. ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

theorem Triangle Angle-Side Relationships Theorem The longest side of a triangle is opposite the largest angle The shortest side of a triangle is opposite the smallest angle. A B C shortest side largest angle longest side smallest angle http://www.mathopenref.com/common/appletframe.html?applet=trianglebigsmall&wid=600&ht=300

example Write the measures for the sides of the triangle in order from least to greatest A B C 111o 46o 23o

You Try Write the measures for the angles of the triangle in order from least to greatest T U 10 V 7 11

You Try J L K

theorem Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. A 1 C B

theorem Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. A 1 C B

example 6 7 3 5 4 2 1

example Write an equation or inequality to describe the relationship between the measures of all angles. ao do co bo

theorem Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A B C A B C A C

theorem Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A B C B A C B C

theorem Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A B C A C B A B http://www.mathopenref.com/common/appletframe.html?applet=triangleinequality2&wid=600&ht=200

example 5, 7, 8 5 + 7 > 8 5 + 8 > 7 7 + 8 > 5 Yes Can a triangle be constructed with sides of the following measures? 5, 7, 8 5 + 7 > 8 5 + 8 > 7 7 + 8 > 5 Yes By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

example Can a triangle be constructed with sides of the following measures? 3, 6, 10 3 + 6 > 10 No

example 11, 15, 21 11 + 15 > 21 15 + 21 > 11 11 + 21 > 15 Yes Can a triangle be constructed with sides of the following measures? 11, 15, 21 11 + 15 > 21 15 + 21 > 11 11 + 21 > 15 Yes By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

example Can a triangle be constructed with sides of the following measures? 4.2, 4.2, 8.4 4.2 + 4.2 > 8.4 No

A triangle cannot be constructed. You Try Can a triangle be constructed with sides of the following measures? 3 in, 3 in , 8 in 3 + 3 > 8 A triangle cannot be constructed. By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

A triangle cannot be constructed. You Try Can a triangle be constructed with sides of the following measures? 6 in, 6 in , 12 in 6 + 6 > 12 A triangle cannot be constructed. By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

A triangle can be constructed. You Try Can a triangle be constructed with sides of the following measures? 9 in, 5 in , 11 in 9 + 5 > 11 A triangle can be constructed. By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

example Solve the inequality.

example x + 8 > 17 x + 17 > 8 8 + 17 > x x > 9 9 < x A triangle has one side of 8 cm and another of 17 cm. Describe the possible lengths of the third side. x + 8 > 17 x + 17 > 8 8 + 17 > x x > 9 9 < x x > -9 25 > x x < 25 17 8 x By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

example x + 11 > 16 x + 16 > 11 11 + 16 > x x > 5 5 < x A triangle has one side of 11 in and another of 16 in. Describe the possible lengths of the third side. x + 11 > 16 x + 16 > 11 11 + 16 > x x > 5 5 < x x > -5 27 > x x < 27 16 11 x By the Triangle Inequality Theorem, the sum of the measures of any two sides must by greater than the third side.

Today’s Assignment p. 298: 1 – 6, 9, 12, 13, 16, 17, 20, 23 – 25, 27, 28 (worksheet in packet)